Question:
Find the average of even numbers from 4 to 936
Correct Answer
470
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 936
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 936 are
4, 6, 8, . . . . 936
After observing the above list of the even numbers from 4 to 936 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 936 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 936
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 936
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the even numbers from 4 to 936
= 4 + 936/2
= 940/2 = 470
Thus, the average of the even numbers from 4 to 936 = 470 Answer
Method (2) to find the average of the even numbers from 4 to 936
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 936 are
4, 6, 8, . . . . 936
The even numbers from 4 to 936 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 936
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the even numbers from 4 to 936
936 = 4 + (n – 1) × 2
⇒ 936 = 4 + 2 n – 2
⇒ 936 = 4 – 2 + 2 n
⇒ 936 = 2 + 2 n
After transposing 2 to LHS
⇒ 936 – 2 = 2 n
⇒ 934 = 2 n
After rearranging the above expression
⇒ 2 n = 934
After transposing 2 to RHS
⇒ n = 934/2
⇒ n = 467
Thus, the number of terms of even numbers from 4 to 936 = 467
This means 936 is the 467th term.
Finding the sum of the given even numbers from 4 to 936
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given even numbers from 4 to 936
= 467/2 (4 + 936)
= 467/2 × 940
= 467 × 940/2
= 438980/2 = 219490
Thus, the sum of all terms of the given even numbers from 4 to 936 = 219490
And, the total number of terms = 467
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 4 to 936
= 219490/467 = 470
Thus, the average of the given even numbers from 4 to 936 = 470 Answer
Similar Questions
(1) Find the average of the first 2011 odd numbers.
(2) Find the average of even numbers from 8 to 502
(3) Find the average of the first 4342 even numbers.
(4) Find the average of even numbers from 6 to 1926
(5) Find the average of the first 2470 even numbers.
(6) What is the average of the first 525 even numbers?
(7) What is the average of the first 533 even numbers?
(8) Find the average of odd numbers from 13 to 567
(9) Find the average of the first 2121 even numbers.
(10) Find the average of the first 2223 even numbers.